Orientational dynamics of weakly inertial axisymmetric particles in steady viscous flows

TL;DR

This paper derives an asymptotic model for the orientation of weakly inertial axisymmetric particles in steady viscous flows, analyzing their dynamics, stability, and distribution under various flow and noise conditions.

Contribution

It introduces a new asymptotic equation of motion for small Stokes number particles, revealing inertia-induced coupling effects and their influence on particle orientation.

Findings

Inertia causes coupling between precession and nutation in particle dynamics.

Prolate particles tend to tumbling, oblate particles tend to log-rolling.

Inertial effects dominate the orientational distribution at high Peclet numbers.

Abstract

The orientational dynamics of weakly inertial axisymmetric particles in a steady flow is investigated. We derive an asymptotic equation of motion for the unit axial vector along the particle symmetry axis, valid for small Stokes number St, and for any axisymmetric particle in any steady linear viscous flow. This reduced dynamics is analysed in two ways, both pertain to the case of a simple shear flow. In this case inertia induces a coupling between precession and nutation. This coupling affects the dynamics of the particle, breaks the degeneracy of the Jeffery orbits, and creates two limiting periodic orbits. We calculate the leading-order Floquet exponents of the limiting periodic orbits and show analytically that prolate objects tend to a tumbling orbit, while oblate objects tend to a log-rolling orbit, in agreement with previous analytical and numerical results. Second, we analyse the role of the limiting orbits when rotational noise is present. We formulate the Fokker-Planck equation describing the orientational distribution of an axisymmetric particle, valid for small St and general Peclet number Pe. Numerical solutions of the Fokker-Planck equation, obtained by means of expansion in spherical harmonics, show that stationary orientational distributions are close to the inertia-free case when Pe St << 1, whereas they are determined by inertial effects, though small, when Pe >> 1/St >> 1.